2021/07/08 by Ryba, Christopher
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2107.03752
A result of Farahat and Higman shows that there is a ``universal'' algebra, FH, interpolating the centres of symmetric group algebras, Z(ℤSn). We explain that this algebra is isomorphic to R ⊗ Λ, where R is the ring of integer-valued polynomials and Λ is the ring of symmetric functions. Moreover, the isomorphism is via ``evaluation at Jucys-Murphy elements'', which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products Γ\wr Sn of a fixed finite group Γ. This involves constructing wreath-product versions RΓ and Λ(Γ_*) of R and Λ, respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, FHΓ, is isomorphic to RΓ⊗ Λ(Γ_*) and use this to compute the p-blocks of wreath products.